168 XIII INTEGRATION mapping gmn: X-> [0,1] such that gmn(x) = 1 for all x e Um and gmn(x) = 0 for all jc e X Un; apply Problem 10(e) to each of the functions gmn. Finally, to show that E has measure 1, show (with the notation of Problem 10) that, for any measure v el, )(x))2 d[jLx(y) = 0 for almost all x e X and all/e <*f(X), by applying Problem 10(d); now let/run through a dense sequence of functions in ^(X).) (b) With respect to the vague topology on M(X) show that, for every measure v e I> ixdv(x) in the sense of (13.10), and that the external points of I are the measures which are ergodic with respect to u (cf. Section 13.10, Problem 9). (c) For every ergodic measure v e I, let Qv (resp. Rv) be the set of points x e Q (resp. x e R) such that px = v. The measure v is concentrated on Rv (13.18). The sets Qv and Rv are called the quasi-ergodic set and the ergodic set corresponding to v. The Qv (resp. Rv) form a partition of Q (resp. R), and we have w(Qv) = Qv and w(Rv) = Rv. Show that, for each closed set F such that w(F) = F, we have either F n Rv = Rv or F n Ry= 0. (d) For every nonempty closed set F such that w(F) = F, show that R n F ^ 0 (consider the points which are regular with respect to u \ F). (e) Let Jt be the monoid generated by lx and u. Show that if Z is any minimal closed orbit with respect to ^ (Section 12.10, Problem 6) we have u(Z) = Z and hence R n Z 9*= 0. Deduce that, for each x e X, the closed orbit O(x) of x with respect to ^ intersects R, and that O(x) n R is a union of ergodic sets Rv. (f) For each x e X, let jit be a measure which is a cluster point of the sequence (pn, x) in M(X). Show that ft(O(jc)) = 1. (g) Show that for a measure (JL e I to be ergodic it is necessary and sufficient that there should exist an ergodic set Rv such that /x(Rv) = 1, and that then we have ju. = v. (Use (b) to prove that the condition is sufficient.) (h) If A; e X is such that O(x) contains only one ergodic set, show that x is an ergodic point (use (f) and (g)). (i) Suppose that I consists of a single measure v0. Then Q = X, and R = D is the support of VQ (use (h)) and is the only minimal closed orbit with respect to M. More- over, for each function /e ^(X), the sequence of numbers - ]T /("*(*)) converges to n k=o \fdvQ uniformly in x e X. (Observe that, for each n, the mapping xt-+pn> x of X into M(X) is continuous with respect to the vague topology, and use (7.5.6).) 12. For every real number p such that 0 < p < +00, and every mapping /: X ->R, put (a) If p > 1, put q = p/(p 1). If/, g are any two mappings of X into R, show that NiCfc) ^ NP(/)N,(0) ("Holder's inequality "). (Show that the set of points fo, t2) e R2 such that fi ^ 0 and r2 ^ 0 and t\lp tŁ/g ^ 1 is convex, and argue as in Section 13.8, Problem 14(c), making use of (13.5.6). Notice that the proof of (13.11.2.2) is a particular case.) (b) If / > 0, g > 0 and p > 1, then Np(/ + #) < Np(/) + Np(#) (" Minkowski's in- equality"). (Same method as (a): consider the set of points (tt, t2) such that rx > 0 and t2 > 0 and t\tp + t\'p > 1.)eparable real Frechet space, K a compact convex subset of E; let u be a